Science (English Medium) Class 12 - CBSE Question Bank Solutions for Mathematics

If y = sin (sin x), prove that `(d^2y)/(dx^2) + tan x dy/dx + y cos^2 x = 0`

If \[A = \begin{bmatrix}1 & 2 \\ 0 & 3\end{bmatrix}\] is written as B + C, where B is a symmetric matrix and C is a skew-symmetric matrix, then B is equal to.

For what value of x, is the matrix \[A = \begin{bmatrix}0 & 1 & - 2 \\ - 1 & 0 & 3 \\ x & - 3 & 0\end{bmatrix}\] a skew-symmetric matrix?

If a matrix A is both symmetric and skew-symmetric, then

The matrix \[\begin{bmatrix}0 & 5 & - 7 \\ - 5 & 0 & 11 \\ 7 & - 11 & 0\end{bmatrix}\] is

If A is a square matrix, then AA is a

If A and B are symmetric matrices, then ABA is

If A = [a_ij] is a square matrix of even order such that a_ij = i² − j², then 

If \[A = \begin{bmatrix}2 & 0 & - 3 \\ 4 & 3 & 1 \\ - 5 & 7 & 2\end{bmatrix}\]  is expressed as the sum of a symmetric and skew-symmetric matrix, then the symmetric matrix is  

If A and B are two matrices of order 3 × m and 3 × n respectively and m = n, then the order of 5A − 2B is 

If A and B are matrices of the same order, then AB^T − BA^T is a 

The matrix  \[A = \begin{bmatrix}0 & - 5 & 8 \\ 5 & 0 & 12 \\ - 8 & - 12 & 0\end{bmatrix}\] is a 

The matrix   \[A = \begin{bmatrix}1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 4\end{bmatrix}\] is

Find the vector and Cartesian equations of the line passing through (1, 2, 3) and parallel to the planes \[\vec{r} \cdot \left( \hat{i}  - \hat{j} + 2 \hat{k}  \right) = 5 \text{ and } \vec{r} \cdot \left( 3 \hat{i} + \hat{j}  + 2 \hat{k} \right) = 6\]

Find the vector and Cartesian forms of the equation of the plane passing through the point (1, 2, −4) and parallel to the lines \[\vec{r} = \left( \hat{i} + 2 \hat{j}  - 4 \hat{k}  \right) + \lambda\left( 2 \hat{i}  + 3 \hat{j}  + 6 \hat{k}  \right)\] and \[\vec{r} = \left( \hat{i}  - 3 \hat{j}  + 5 \hat{k}  \right) + \mu\left( \hat{i}  + \hat{j}  - \hat{k} \right)\] Also, find the distance of the point (9, −8, −10) from the plane thus obtained.  

Find the vector equation of the line passing through (1, 2, 3) and parallel to the planes  \[\vec{r} \cdot \left( \hat{i}  - \hat{j}  + 2 \hat{k}  \right) = 5 \text{ and } \vec{r} \cdot \left( 3 \hat{i}  + \hat{j}  + \hat{k}  \right) = 6 .\]

Find the vector and cartesian equations of the plane passing throuh the points (2,5,- 3), (-2, - 3,5) and (5,3,-3). Also, find the point of intersection of this plane with the line passing through points (3, 1, 5) and (–1, –3, –1).

Find the equation of the plane passing through the intersection of the planes `vec(r) .(hat(i) + hat(j) + hat(k)) = 1"and" vec(r) . (2 hat(i) + 3hat(j) - hat(k)) +4 = 0 `and parallel to x-axis. Hence, find the distance of the plane from x-axis.